\section{UEP Methods}

\begin{frame}{Non-expanding Windows}
Definition:
\begin{itemize}
\item{Probabilistic layer decision rule: $\boldsymbol \Gamma$}
\item{Layer sizes: $k_1,k_2,\hdots,k_i$}
\end{itemize}
\begin{center}
\begin{figure}[h!]
\centering
\resizebox{0.9\linewidth}{!}{
	\begin{tikzpicture}[>=stealth',shorten >=1pt,auto, semithick]
	\tikzstyle{every state}=[fill=white,text=black,, minimum height=0.7cm, minimum width=3cm, node distance=3cm,font=\normalfont]
	
		\node[state, rectangle, fill=black!30] (L11) {Layer 1 (L1)};
		\node[state, rectangle, right of=L11](L12){};
		\node[state, rectangle, right of=L12](L13){};
		\node[state, rectangle, right of=L13](L14){};

		\node[state, rectangle, below of=L13,node distance=0.7cm](L23){};
		\node[state, rectangle, right of=L23,node distance=3cm](L24){};
		\node[state, rectangle, below of=L11,node distance=0.7cm](L21){};
		
		\node[state, rectangle, below of=L11,node distance=1.4cm,yshift=-0cm](L31){};
		\node[state, rectangle, right of=L31,node distance=3cm](L32){};
		\node[state, rectangle, right of=L32,node distance=3cm](L33){$\hdots$};
		\node[state, rectangle, right of=L33,node distance=3cm](L34){};

		\node[state, rectangle, left of=L23, fill=black!30] (L22) {Layer 2 (L2)};
		\node[state, rectangle, below of=L34, fill=black!30,node distance=0.7cm] (L44) {Layer \textit{i} (Li)};
		\node[state, rectangle, below of=L33,node distance=0.7cm](L43){};
		\node[state, rectangle, below of=L32,node distance=0.7cm](L42){};
		\node[state, rectangle, below of=L31,node distance=0.7cm](L41){};
		
		\draw [decorate,thick,decoration={brace, amplitude=10pt},yshift=20pt] (L11.north west) -- (L12.north west) node[midway,above,yshift=7pt]{$k_1$};		
		\draw [decorate,thick,decoration={brace, amplitude=10pt},yshift=20pt] (L12.north west) -- (L13.north west) node[midway,above,yshift=7pt]{$k_2$};
		\draw [decorate,thick,decoration={brace, amplitude=10pt},yshift=40pt] (L13.north west) -- (L14.north west) node[midway,above,yshift=10pt]{$\mathbb{\hdots}$};
		\draw [decorate,thick,decoration={brace, amplitude=10pt},yshift=20pt] (L14.north west) -- (L14.north east) node[midway,above,yshift=7pt]{$k_i$};
		\draw [decorate,thick,decoration={brace, mirror, amplitude=10pt},yshift=-20pt] (L41.south west) -- (L44.south east) node[midway,below,yshift=-7pt]{Source block};
				
		\node[left of=L11,xshift=-1.5cm](){$\boldsymbol \Gamma_1$};
		\node[left of=L21,xshift=-1.5cm](){$\boldsymbol \Gamma_2$};
		\node[left of=L31,xshift=-1.5cm](){$\vdots$};
		\node[left of=L41,xshift=-1.5cm](){$\boldsymbol \Gamma_i$};
	\end{tikzpicture}}
\caption{3.11, Page 39}
\end{figure}
\end{center}
\end{frame}

\begin{frame}{Expanding Windows}
Definition:
\begin{itemize}
\item{Probabilistic layer decision rule: $\boldsymbol \Gamma$}
\item{Layer sizes: $K_1,K_2,\hdots,K_i$}
\end{itemize}
\begin{center}
\begin{figure}[h!]
\centering
\resizebox{0.9\linewidth}{!}{
	\begin{tikzpicture}[>=stealth',shorten >=1pt,auto, semithick]
	\tikzstyle{every state}=[fill=white,text=black,, minimum height=0.7cm, minimum width=3cm, node distance=3cm]
		\node[state, rectangle, fill=black!30] (L11) {Layer 1 (L1)};
		\node[state, rectangle, right of=L11](L12){};
		\node[state, rectangle, right of=L12](L13){};
		\node[state, rectangle, right of=L13](L14){};

		\node[state, rectangle, below of=L13,node distance=0.7cm](L23){};
		\node[state, rectangle, right of=L23,node distance=3cm](L24){};

		\node[state, rectangle, below of=L11,node distance=1.4cm,yshift=-0cm](L31){$\hdots$};
		\node[state, rectangle, right of=L31,node distance=3cm](L32){$\hdots$};
		\node[state, rectangle, right of=L32,node distance=3cm](L33){$\hdots$};
		\node[state, rectangle, right of=L33,node distance=3cm](L34){};

		\node[state, rectangle,minimum width=6cm, left of=L23, fill=black!30,node distance=4.5cm] (L22) {Layer 2 (L2)};
		\node[state, rectangle, below of=L31, fill=black!30,node distance=0.7cm, minimum width=12cm, xshift=4.5cm] (L33) {Layer \textit{i} (Li)};		
		
		\draw [decorate,thick,decoration={brace, amplitude=10pt},yshift=20pt] (L11.north west) -- (L12.north west) node[midway,above,yshift=7pt]{$k_1$};		
		\draw [decorate,thick,decoration={brace, amplitude=10pt},yshift=20pt] (L12.north west) -- (L13.north west) node[midway,above,yshift=7pt]{$k_2$};
		\draw [decorate,thick,decoration={brace, amplitude=10pt},yshift=40pt] (L13.north west) -- (L14.north west) node[midway,above,yshift=10pt]{$\mathbb{\hdots}$};
		\draw [decorate,thick,decoration={brace, amplitude=10pt},yshift=20pt] (L14.north west) -- (L14.north east) node[midway,above,yshift=7pt]{$k_i$};
		\draw [decorate,thick,decoration={brace, mirror, amplitude=10pt},yshift=-20pt] (L33.south west) -- (L33.south east) node[midway,below,yshift=-7pt]{Source block};
				
		\node[left of=L11,xshift=-1.5cm](){$\boldsymbol \Gamma_1$};
		\node[left of=L22,xshift=-3.0cm](){$\boldsymbol \Gamma_2$};
		\node[left of=L31,xshift=-1.5cm](){$\vdots$};
		\node[left of=L33,xshift=-6cm](){$\boldsymbol \Gamma_i$};
	\end{tikzpicture}}
\caption{3.15, Page 45}
\end{figure}
\end{center}
\end{frame}

\begin{frame}{Evaluation and Comparison of UEP Methods}
Non-expanding Windows:
\begin{itemize}
\item{$k_1=32,k_2=64$}
%\item{$\boldsymbol \Gamma_1=\{0.3 , 0.4 , 0.5\}$}
%\item{$q=\{2,256\}$}
\end{itemize}
$ $\\
Expanding Windows:
\begin{itemize}
\item{$K_1=32,K_2=96$}
%\item{$\boldsymbol \Gamma_1=\{0.3 , 0.4 , 0.5\}$}
%\item{$q=\{2,256\}$}
\end{itemize}
$ $\\
Parameters:
\begin{itemize}
\item{$\boldsymbol \Gamma_1=\{0.3 , 0.4 , 0.5\}$}
\item{$q=\{2,256\}$}
\end{itemize}
$ $\\
Performance measure: When can data be decoded?
\end{frame}

\begin{frame}{Evaluation and Comparison of UEP Methods}
$\Gamma_1=0.3$
\vspace{-0.6cm}
\begin{figure} \centering
\caption{Fig: 3.14(a) on Page 43, 3.19(a) on Page 49}
\vspace{-0.6cm}
\subfloat{\includegraphics[width=0.8\textwidth]{../rapport/figs/uep_new_analytic_g1_30_g2_70}}\\
\vspace{-0.45cm}
\subfloat{\includegraphics[width=0.8\textwidth]{../rapport/figs/uep_ew_analytic_g1_03_g2_07}}
\end{figure}
\end{frame}

\begin{frame}{Evaluation and Comparison of UEP Methods}
$\Gamma_1=0.4$
\vspace{-0.6cm}
\begin{figure} \centering
\caption{Fig: 3.14(b) on Page 43, 3.19(b) on Page 49}
\vspace{-0.6cm}
\subfloat{\includegraphics[width=0.8\textwidth]{../rapport/figs/uep_new_analytic_g1_40_g2_60}}\\
\vspace{-0.45cm}
\subfloat{\includegraphics[width=0.8\textwidth]{../rapport/figs/uep_ew_analytic_g1_04_g2_06}}
\end{figure}
\end{frame}

\begin{frame}{Evaluation and Comparison of UEP Methods}
$\Gamma_1=0.5$
\vspace{-0.6cm}
\begin{figure} \centering
\caption{Fig: 3.14(c) on Page 43, 3.19(c) on Page 49}
\vspace{-0.6cm}
\subfloat{\includegraphics[width=0.8\textwidth]{../rapport/figs/uep_new_analytic_g1_50_g2_50}}\\
\vspace{-0.45cm}
\subfloat{\includegraphics[width=0.8\textwidth]{../rapport/figs/uep_ew_analytic_g1_05_g2_05}}
\end{figure}
\end{frame}

\begin{frame}{Evaluation and Comparison of UEP Methods}
Summary:
\begin{itemize}
\item{Decoding probability}
\item{Expanding Windows superior to Non-expanding Windows}
\end{itemize}
$ $\\
Open Questions:
\begin{itemize}
\item{Choosing $\boldsymbol \Gamma$ (Hands-on later)}
\item{Choosing the number of layers (Hands-on later)}
\end{itemize}
\end{frame}






































